This invention relates generally to inertial navigation systems and more specifically to determining gravity in such systems.
The velocity V of interest in navigating a vehicle relative to the earth is defined by the equation ##EQU1## where ##EQU2## is the rate of change of the vehicle's velocity relative to the earth expressed in a NAV (N) frame of reference (local-level with origin fixed at the center of the earth), a.sub.s is the specific-force acceleration experienced by the inertial navigation system on board the vehicle, g is gravity, .PHI. is the rotation rate of an earth-fixed frame of reference relative to an inertial frame (i.e. earth's rotation rate vector), R is the position vector of the vehicle from the center of the earth, and .omega. is the rotation rate of the local-level frame relative to the inertial frame. In order to integrate ##EQU3## and obtain V, an accurate expression for g is required.
The so-called normal gravity potential .PHI. (the most accurate gravity model presently available) is given in terms of ellipsoidal coordinates (.mu.,.beta.,.lambda.) as ##EQU4## and ##EQU5## where G.sub.m is the earth's gravitational constant, .alpha. is the semi-major axis of WGS-84 ellipsoid (DMA Technical Report, Department of Defense WGS-84, TR 8350.2), b is the semi-minor axis of WGS-84 ellipsoid, (e,n,u) are the vehicle coordinates in the NAV (N) frame (e-east, n-north, u-up), and .OMEGA. is the earth rotation rate.
Given the vehicle's location in geodetic coordinates (.phi., .lambda., h), the normal gravity vector expressed in the ECEF (earth-centered, earth-fixed) frame is given by ##EQU6## where ##EQU7## is the transformation matrix from the ellipsoidal (U) frame to the ECEF (E) frame, and g.sup.U, the normal gravity vector, is expressed in ellipsoidal coordinates as ##EQU8## Note that u=b, q=q.sub.0, .nu.=a, and g.sub..beta. =0 when h=0.
The normal gravity vector expressed in NAV (N) frame coordinates is given by ##EQU9## where ##EQU10##
To determine the normal gravity vector at the vehicle's location expressed in NAV-frame coordinates, one first transforms to geodetic coordinates, then to ECEF coordinates, and finally to ellipsoidal coordinates. The geodetic-to-ECEF transformation is defined by the equations EQU x=(N+h) cos .phi. cos .lambda. EQU y=(N+h) cos .phi. sin .lambda. EQU z=(Nb.sup.2 /a.sup.2 +h) sin .phi. (17)
where ##EQU11## The transformation to ellipsoidal coordinates is defined by the equations ##EQU12## where EQU r.sup.2 =x.sup.2 +y.sup.2 +z.sup.2 (20)
When the vehicle's position in ellipsoidal coordinates has been determined, then the normal gravity vector in ellipsoidal coordinates can be calculated. The final step is to transform the normal gravity vector from ellipsoidal coordinates to NAV-frame coordinates using the equations presented above.
To simplify the process of determining the gravity vector, the so-called J.sub.2 gravity model is utilized in present-day inertial navigation systems. The J.sub.2 gravity model can be expressed in rectangular ECEF coordinates and thereby greatly reduces the computational load associated with determining the gravity vector at a vehicle's location. Unfortunately, the cost of this reduction is a reduction in accuracy of the gravity vector.
A need exists for a gravity-determining procedure which provides the accuracy of the normal model and can be implemented with currently-available inertial navigation system processors.